NA TURE 



145 



THURSDAY, DECEMBER 13, 1894. 



DILETTANTISM AND STATISTICS. 

 The Gro-i'th of St. Louis Children. By William Town- 

 send Porter. Vol. vi. No 12 : Transactions of the 

 Academy of Science of St. Louis. 



THE anthropometrical researches of Mr. Francis 

 Gallon have had a very widespread influence, and, 

 under his inspiration, a large mass of material has been, 

 and is being collected, which cannot fail to be of 

 service in elucidating a number of knotty problems. In 

 particular, very comprehensive measurements of children 

 have been made in America by Bowditch, Sargant, and 

 Porter, the results being tabulated by aid of Mr. Galton's 

 method of percentiles. 



It is not only in the form of statistics of measurements 

 on man, but also of measurements on the lower animals 

 and on plants, that material for the study of variation 

 and correlation is accumulating with almos'. alarm- 

 ing rapidity. VVe say with almost alarming rapidity, 

 for not only is theory lagging somewhat behind the 

 needs of statistical experience, but, what is far 

 worse, many collectors have no real insight into the 

 theory which does e.xist. They are tabulating results in 

 a form which will be of no permanent service, or neglect- 

 ing to publish the very details which alone would enable 

 us to test any development of statistical theory. The 

 ignorance of the elements of mathematics, to say nothing 

 of the theory of chance, is, indeed, singularly character- 

 istic of many investigators, who think statistics can be 

 handled, and conclusions drawn from them, without the 

 least preliminary training. For e.xample, in a recent 

 paper on variation in the flowers of buttercups and 

 clover, by Herr de \'ries, we are introduced to our old 

 friend the normal curve of errors as a definite sym- 

 metrical polygon ; no attempt is made to fit it by aid of 

 the probable error of the observations, but a particular 

 form of it, apparently selected at random., is plumped 

 down on the top of another polygon representing the 

 observations. The odds against the thus selected 

 theoretical polygon (curiously called the " Galton Curve ") 

 expressing the real distribution of variation, are, it is need- 

 less to say, many thousands to one. Still more curiously, 

 sets of observations giving theoretically very good fre- 

 quency curves of the type which occurs in infantile 

 mortality, the valuation of houses, &c., are termed 

 " Half-Galton-Curves," although the type, so far from 

 being represented by the half of the normal curve of 

 errors, corresponds to a curve asymptotic to the ordinate 

 of maximum frequency ! We might pass these eccen- 

 tricities by, were not statistics thus handled used to 

 support some vague theory of heredity, or to question 

 some principle of variation. 



Mr. Porter is not quite so wild in theory as Herr 

 de Vries, but for him also the normal curve of errors 

 appears to be a polygon, and he tells us, on p. 277, after 

 a table of its ordinates — which are quoted from Thoma, 

 and given only to the units, and not to decimals — " that 

 there is a limit beyond wkici; no deviation occurs." It is, 

 perhaps, needless to say that if we start only with a know- 

 ledge of the theory of chance, such as may be found in 

 NO. 1311, VOL. 51J 



works like those of Thoma and Stieda, we are hardly 

 likely to avoid bad theoretical blunders. Mr. Porter 

 is no worse than many writers of memoirs in the Toiirnal 

 of the Royal Statistical Society; but the time has come 

 when statistics, as well as archaeology, must be taken 

 out of the hands of the dilettanti, and when a man, if 

 he takes upon himself to publish statistical researches, 

 must, like the physicist, show his credentials in the form 

 of a fair mathematical training. 



Unfortunately, Mr. Galton's method of percentiles, 

 useful as it is in the right place, is now acting as a 

 distinct check to statistical theory in a somewhat un- 

 expected manner. Before that theory was disseminated 

 the dilettante statistician was compelled to publish his raw 

 material, and his remarks on it did not impair its value 

 to the trained scientist. Now, however, the percentile 

 method of dealing with statistics seems so intelligible, 

 that the non-mathematical anthropologist or physiologist 

 grasps it at once, reserves his raw material, and pub- 

 lishes tables of percentiles. It is hard to conceive any- 

 thing more disastrous for true statistical progress. 



The reason for this is easily explained. As is well 

 known, any normal curve of errors can be reduced to any 

 other by uniform stretches (or squeezes) parallel to its 

 base and its axis. The area of any portion of any normal 

 curve up to a deviation of given magnitude cannot be 

 given by a finite series ; it can by stretching be reduced 

 to an integral, the values of which are tabulated, but 

 which is not itself finitely expressible for an indefinite 

 value of the deviation. The tables formed of this in- 

 tegral, however, enable us to pass from selected percen- 

 tiles to the constants, and so to the form of the frequency 

 curve. Now, innumerable frequen cy curves in biological, 

 just as in economical statistics, are »<•/ of the normal type. 

 .•\ generalised frequency curve can be theoretically ob- 

 tained fitting closely the observations in many, perhaps in 

 all such cases, but owing to its asymmetry or skewness, its 

 limited range and other features, it neither possesses the 

 stretch property of the normal curve, nor can its areas 

 be expressed by tables of one entry. The tables required 

 are, according to the type, of two or three entries, and 

 such tables are not likely to be prepared for a long time 

 to come, and if prepared, would be of little service except 

 for the case when the statistics are given in percentiles. In 

 other words, the use of percentiles may suffice to demon- 

 strate the skewness of the statistics, but at once precludes 

 the use of any theory higher than that of the normal curve, 

 by which the skewness could be accounted for. This is the 

 fundamental defect of Mr. Porter's labours. He gives us a 

 most extensive system of measurement on boys and girls 

 from six to twenty years of age— ample material, if properly 

 dealt with, to provide solutions for innumerable problems 

 in correlation and selection with age. The material, how- 

 ever, is only given in the form of percentiles or in dia- 

 grams of the "ogive" curve corresponding to the integral 

 of the frequency curve. These diagrams fully confirm 

 the results of Bowditch— Ma/ the statistics are skew, and 

 that the degree of sxe'u'ness varies -ujith the age. But all 

 means of scientifically treating this skewness disappear, 

 because we have only percentile results. The skewness 

 of theoriginal frequency curve cannot be deduced from nine 

 or ten points on a curve, the equation of which is only 

 expressible by an unintegratable form containing the con- 



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